Question 120
We say that a simple periodic decimal fraction $f$ has the reduced length equal to $n$ (where $n$ is a positive integer) if $f$ has a $n$-digit period and $f$ cannot be represented as a simple periodic decimal fraction with a period having less than $n$ digits. For instance, $0.(223)$ has the reduced length 3, while $0.(2323)$ has the reduced length 2, as $0.(2323) = 0.(23)$.
a) Prove that $f = 0.(2) \cdot 0.(3)$ is a simple periodic fraction with reduced length 3.
b) Do there exist two simple periodic fractions with reduced length 1, such that their product has reduced length also 1?
c) Do there exist two simple periodic fractions with reduced length 3, such that their product has the reduced length also 3?
a) Prove that $f = 0.(2) \cdot 0.(3)$ is a simple periodic fraction with reduced length 3.
b) Do there exist two simple periodic fractions with reduced length 1, such that their product has reduced length also 1?
c) Do there exist two simple periodic fractions with reduced length 3, such that their product has the reduced length also 3?